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Weak Convergence of Self-normalized Partial Sums Processes

  • Miklós Csörgő
  • Zhishui HuEmail author
Part of the Fields Institute Communications book series (FIC, volume 76)

Abstract

Let {X, X n , n ≥ 1} be a sequence of independent and identically distributed non-degenerate random variables. Put \(S_{0} = 0,\ S_{n} =\sum _{ i=1}^{n}X_{i}\) and \(V _{n}^{2} =\sum _{ i=1}^{n}X_{i}^{2},\ n \geq 1.\) A weak convergence theorem is established for the self-normalized partial sums processes \(\{S_{[int]} /V _{n},0 \leq t \leq 1\}\) when X belongs to the domain of attraction of a stable law with index α ∈ (0, 2]. The respective limiting distributions of the random variables \(\max _{1\leq i\leq n}\vert X_{i}\vert /S_{n}\) and \(\max _{1\leq i\leq n}\vert X_{i}\vert /V _{n}\) are also obtained under the same condition.

Notes

Acknowledgements

We wish to thank two referees for their careful reading of our manuscript. The present version reflects their much appreciated remarks and suggestions. In particular, we thank them for calling our attention to the newly added reference Kallenberg [19], and for advising us that the proof of our Theorem 2.1 needs to be done more carefully, taking into account the remarks made in this regard. The present revised version of the proof of our Theorem 2.1 is done accordingly, with our sincere thanks attached herewith.

This research was supported by an NSERC Canada Discovery Grant of Miklós Csörgő at Carleton University and, partially, also by NSFC(No.10801122), the Fundamental Research Funds for the Central Universities, obtained by Zhishui Hu.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsCarleton UniversityOttawaCanada
  2. 2.Department of Statistics and Finance, School of ManagementUniversity of Science and Technology of ChinaHefei, AnhuiChina

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